A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition

Abstract

A continuous interior penalty finite element method (CIP-FEM) is proposed to solve high-frequency Helmholtz scattering problem by an impenetrable obstacle in two dimensions. To formulate the problem on a bounded domain, a Dirichlet-to-Neumann (DtN) boundary condition is proposed on the outer boundary by truncating the Fourier series of the original DtN mapping into finite terms. Assuming the trunca-tion order N ≥ kR, where k is the wave number and R is the radius of the outer bound-ary, then the H^j -stabilities, j = 0,1,2, are established for both original and dual problems, with explicit and sharp estimates of the upper bounds with respect to k. More-over, we prove that, when N ≥ λkR for some λ > 1, the solution to the DtN-truncation problem converges exponentially to the original scattering problem as N increases. Under the condition that k³h² is sufficiently small, we prove that the preasymptotic error estimates for the linear CIP-FEM as well as the linear FEM are C₁kh+C₂k³h². Numerical experiments are presented to validate the theoretical results.

Keywords

Helmholtz equation / high-frequency / DtN operator / CIP-FEM / wave-number-explicit estimates